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    "## 3.5 Superconducting circuits\n",
    "\n",
    "One of the greatest technological advancements in the 20th century was the CMOS (complementary\n",
    "metal–oxide–semiconductor) fabrication process. This is a method to fabricate integrated circuits by\n",
    "stacking layers of materials on [wafers](https://en.wikipedia.org/wiki/Wafer_(electronics)) repeatedly at large scale. Different materials can be layered onto\n",
    "wafers at nanometer or micrometer precision to form circuit structures. The wafers are then cut into chips\n",
    "to be mounted to a larger system (see \"dilution refrigerator\"). Quantum dots, in session 3.2, are also\n",
    "fabricated this way.\n",
    "\n",
    "Just like atomic levels where electrons occupy in an atom at different energies, we can create a\n",
    "circuit on chip that behaves like an artificial atom – it has different energy levels the system can be at.\n",
    "Now, if we can \"isolate\" the lowest two levels and assign them as states |0⟩ and |1⟩, perhaps we will be\n",
    "able to make a qubit using these two levels. While a natural atom's energy levels are difficult to control,\n",
    "an artificial atom created based on our design could be much more user-friendly.\n",
    "\n",
    "![text](image/LC_circuit.png) \n",
    "\n",
    "It turns out that we can achieve this just by using an inductor-capacitor (LC) circuit. If you are\n",
    "familiar with a generic LC circuit, you would recall that it behaves like a harmonic oscillator. The energy of\n",
    "the circuit is a parabolic function of frequency. This is a classical behavior. But if we make the circuit small\n",
    "enough and cool the circuit down to sub-Kelvin, we enter the quantum regime. (The specific criteria are:\n",
    "effective length of the circuit is smaller than the electron scattering length in the circuit; and the\n",
    "temperature is low enough. How cold is low enough? $kT < \\hbar\\omega$, where k is the Boltzmann constant, T is\n",
    "the temperature and $\\omega = \\sqrt{LC}$ is the natural frequency of the circuit. Typically, with small circuits we\n",
    "make today, the temperature could be below liquid Helium temperature at 4K.)\n",
    "\n",
    "![text](image/LC_energies.png) \n",
    "\n",
    "In the quantum regime, the energy of the circuit splits into discrete levels, which are in\n",
    "superposition. They are equally spaced by $\\hbar\\omega$. We could define the bottom two energy levels to be our\n",
    "|0⟩ and |1⟩, but it would not be very useful because the energy required to change between |0⟩ and |1⟩\n",
    "could as well induce transitions between other adjacent states. We must isolate the levels.\n",
    "\n",
    "This problem can be solved by using a Josephson junction, a special type of inductor, instead of a\n",
    "normal inductor. What it does here is to change the potential from a parabolic to a sinusoidal shape.\n",
    "Interestingly, we can find this behavior in other systems, such as in a pendulum. Recall in _Physics insert –\n",
    "artificial atom_ in session 3.2, the Hamiltonian of a superconducting circuit appears to have a very similar\n",
    "form as the Hamiltonian of a pendulum. We can change the shape of a pendulum's energy function by\n",
    "tweaking the value of gravitational acceleration, g, for example moving from Earth to Mars. With an\n",
    "appropriate value of g, the potential of a pendulum can be a sinusoidal function, just as a superconducting\n",
    "circuit with a Josephson junction can make the potential change from a parabolic to a sinusoidal shape.\n",
    "\n",
    "![text](image/LC_quantum.png) \n",
    "\n",
    "Now that the energy levels are no longer equally spaced, we can use the bottom two levels as our\n",
    "|0⟩ and |1⟩ states. We can fabricate many qubits next to each other to enable entanglement. To control\n",
    "the states, i.e. to apply gates to them, we also use electromagnetic fields like in previous cases. However,\n",
    "instead of shining light or applying external fields, we send microwave signals through the circuits on chip.\n",
    "The sizes of the circuits are comparable with microwave wavelengths. Therefore, we can directly make\n",
    "[transmission lines](https://en.wikipedia.org/wiki/Transmission_line) on chip to guide the wave to the qubits.\n",
    "\n",
    "\n",
    ">_Physics insert – superconductors_ ----------------------------------------------------------------------\n",
    ">\n",
    ">In a conducting solid, the ions in the solid form a lattice. Electrons that are unbound by\n",
    ">ions, often called free electrons, flow in the material, conducting electricity. As\n",
    ">electrons flow through the material, they bump into the lattice and get scattered,\n",
    ">dissipating energy. This is what's causing resistance in a conductor.\n",
    ">\n",
    ">![text](image/conductor.png) \n",
    ">\n",
    ">Superconductors are conductors with little electrical resistance. This means\n",
    ">electrons in superconductors bump into the ion lattice much less than in normal\n",
    ">conductors. How does this happen?\n",
    ">\n",
    ">![text](image/resistance.png) \n",
    ">\n",
    ">(The curvatures are arbitrary and are only for illustration purposes. Not all conductors\n",
    ">have linear resistance to temperature and superconductors do not have the exact\n",
    ">resistance curve as drawn.)\n",
    ">\n",
    ">We know that ions are positively charged, and electrons are negatively charged.\n",
    ">As an electron flows through a lattice of ions, it perturbs the lattice because the ions\n",
    ">around the electron move toward it due to the attractive forces between positive and\n",
    ">negative charges. If another electron comes along, instead of experiencing a repulsion\n",
    ">from the first electron, it experiences an attraction towards this local disturbance.\n",
    ">These two electrons are then bound together, forming a pair. This is called a Cooper\n",
    ">pair (named after physicist Leon Copper). The net effect of this many-body interaction\n",
    ">is that all the conducting electrons form Cooper pairs which moves inside the lattice\n",
    ">together without colliding with the ion lattice.\n",
    ">\n",
    ">![text](image/superconductor.png) \n",
    ">\n",
    ">Superconductors are materials with such conditions met. They behave like\n",
    ">normal conductors, where electrons bump into ion lattices, until temperature is cooled\n",
    ">below a certain critical point, $T_c$ , when Cooper pairs are formed. Not all materials have\n",
    ">a superconducting state. The search for relatively high-temperature superconductors\n",
    ">(high $T_c$ ) is an ongoing effort. The higher the critical temperature, the closer it is to\n",
    ">room temperature, which means it is easier for us to use in everyday life.\n",
    ">\n",
    ">![text](image/Cooper_pair.png) \n",
    ">\n",
    ">_Physics insert – Josephson junction_ -------------------------------------------------------------------\n",
    ">\n",
    ">As described above, the behaviors of an electron in a superconductor correlate to all\n",
    ">other electrons. In other words, the electrons behave coherently. Mathematically, we\n",
    ">can find that Cooper pairs can be described by the same wavefunction with a constant\n",
    ">phase difference (see session 1.1 Physics insert - wavefunction). This contrasts with\n",
    ">regular conductors where all the electrons have different wavefunctions with\n",
    ">independent phases – they are incoherent. This is analogues to coherent photons from\n",
    ">a laser verses incoherent photons from other types of light source.\n",
    ">\n",
    ">![text](image/coherence.png) \n",
    ">\n",
    ">The significance of having coherent Cooper pair phases is that the wavefunctions\n",
    ">between two superconductors can interfere. When two superconductors with phases\n",
    ">$\\phi_1$ and $\\phi_2$ are brought right next to each other, they can form a new superconductor\n",
    ">with a single wavefunction with a new phase $\\phi_3$.\n",
    ">\n",
    ">![text](image/two_superconductors.png) \n",
    ">\n",
    ">But if they are separated by a small barrier, quantum tunneling of wavefunctions\n",
    ">happens and they interfere with a phase difference\n",
    ">\n",
    ">$\\theta = \\phi_1 - \\phi_2$. \n",
    ">\n",
    ">This phase difference causes a supercurrent to flow through the junction with value\n",
    ">\n",
    ">$I_S = I_C \\sin\\theta$\n",
    ">\n",
    ">where \n",
    ">\n",
    ">$I_C$\n",
    ">\n",
    ">is called the critical current, which is the maximum value of supercurrent that\n",
    ">can flow through the junction. This was discovered by Brian Josephson in early 1960s.\n",
    ">Experimentally a Josephson junction is achieved by sandwiching an insulator in\n",
    ">between two superconductors.\n",
    ">\n",
    ">![text](image/josephson_junction.png) \n",
    ">\n",
    ">As this effect is associated with the kinetics of electrons, an inductance can be\n",
    ">derived for the Josephson junctions.\n",
    ">\n",
    ">$L_j = \\frac{L_0}{\\cos\\theta} = \\frac{\\Phi_0}{2 \\pi I_C\\cos\\theta}$\n",
    ">\n",
    ">where\n",
    "> \n",
    ">$\\Phi_0$ is the magnetic flux quantum $\\frac{\\hbar}{2\\epsilon}.$ \n",
    ">\n",
    ">That is why the energy of the LC circuit shown earlier in the chapter is a function of the phase difference \n",
    ">\n",
    ">$\\theta$.\n",
    ">\n",
    ">_Physics insert – low-temperature experiments_ -----------------------------------------------------\n",
    ">\n",
    ">Most of the methods of building quantum computing hardware, from using\n",
    ">semiconducting circuits to making quantum dots to building topological qubits, which\n",
    ">will be discussed later, require the setup to be cooled to very low temperatures (milli-\n",
    ">Kelvin range). This is because temperature gives energy to everything in the system,\n",
    ">which makes the particles move around. We already saw in the case of conductors,\n",
    ">electrons bumping into atom lattice causes electrical resistance. The higher the\n",
    ">temperature, the more vibration and agitation the atoms have, making electrons more\n",
    ">likely to bump into them. When we investigate quantum properties of materials, this\n",
    ">kind of vibrations and agitations of particles result in noise that masks quantum\n",
    ">mechanical signals. When it comes to building a quantum computer requiring delicate\n",
    ">interactions between particles, temperature is our enemy.\n",
    ">\n",
    ">![text](image/heating_noise.png) \n",
    ">\n",
    ">A special cooling unit known as a dilution refrigerator, is built to cool materials\n",
    ">down to milli-Kelvin range. The \"coolant\" used in a dilution refrigerator is a mixture of\n",
    ">Helium-3 and Helium-4 isotopes and they are cooled in stages. First the cryostat is\n",
    ">immersed from room temperature into liquid nitrogen which is naturally at 77 K, inside\n",
    ">which there is a liquid Helium-4 bath at 4.2 K. Then He-3 gas is added into the cryostat\n",
    ">getting precooled to 4.2 K under pressure. The He-3 then enters a vacuum chamber\n",
    ">that is cooled to about 1 K by pumping He-4 bath. This makes the He-3 undergo phase\n",
    ">transition from gas to liquid. Then He-3 enters a still filled with He-4 superfluid. The\n",
    ">pressure in the still is pumped low which keeps superfluid He-4 at a few hundred mK.\n",
    ">The He-3 then enters a mixing chamber of a He-3 and He-4 mixture. The upper layer is\n",
    ">He-3 rich, which is a concentrated phase for He-3, while the lower layer is a diluted\n",
    ">phase for He-3. This phase transition is endothermic, meaning it absorbs heat from its\n",
    ">environment. The quantum computing setup is at the bottom of this mixing chamber,\n",
    ">which is the coolest area of the cryostat at typically 10-100 mK. Cryostat technology\n",
    ">has been an advancing field due to the many researches on quantum materials. The\n",
    ">description here is a typical dilution refrigerator but note there are many other ways\n",
    ">to achieve extremely low temperature.\n",
    ">![text](image/dilution_fridge.png) \n",
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